Mr Currie died in his sleep at his home in Derrymullen, Co Kildare, on Tuesday. He had recently celebrated his 82nd birthday.
Mr Currie, who also helped to found the nationalist SDLP party and was elected to parliament on both sides of the Irish border, was described by Tanaiste Leo Varadkar as “one of the outstanding politicians of his generation”.
Mr Currie was born in Co Tyrone, the eldest of 11 children.
His decision to squat at a council house in Caledon in June 1968 is widely seen as the beginning of the civil rights movement, which challenged inequality and discrimination against Catholics.
He went on to create the SDLP along with John Hume and Gerry Fitt in 1970.
In 1989, he won a seat in Dublin West for Fine Gael and pursued a career as TD and minister in the Republic until he retired in 2002.
A family statement said: “The Currie family is heartbroken to announce the death of Austin Currie.
“Austin was married to Annita for 53 years. They were a formidable team whose love for each other and their family saw them through some of the worst times in Northern Ireland’s recent history.
“He is survived by his children Estelle, Caitriona, Dualta, Austin and Emer, their partners and 13 grandchildren.”
The family statement continued: “Our Daddy was wise, brave and loving and we thank him for the values that he lived by and instilled in us.
“He was our guiding star who put the principles of peace, social justice and equality first.
“From Edendork in county Tyrone to the bog of Allen, Daddy was most at home with his beloved Annita and his family, surrounded by newspapers and grandchildren.
“We will miss him deeply.”
SDLP leader Colum Eastwood said Mr Currie was a “titan” of the civil rights movement.
He added: “His housing protest in Caledon in 1968 was one of the key sparks for the civil rights campaign that followed and he spoke for a generation of young nationalists when he refused to allow his constituents to be treated as second class citizens anymore.
“His radical activism led him to join together with other young leaders and together they formed our party on the principles of a shared society where everyone got a fair shot at life, something so many of their contemporaries had been denied.
“Each time we lose a political giant like Austin we lose a piece of our history.
“While moments like this bring us great sadness, it also gives us the opportunity to celebrate the man and the huge contribution he made to politics in both the North and South of our island.”
Mr Varadkar said: “I am deeply saddened to hear of the death of Austin Currie, and extend my sympathies to his family.
“A pioneer of the civil rights movement in Northern Ireland, Austin was one of the outstanding politicians of his generation, highlighting discrimination against nationalists in issues like housing with a famous sit-in protest at Caledon.
He added: “I knew Austin as a brave, courageous and principled man. He was blessed with extensive political insight and boundless humanity.
“Above all, he cared most about bringing peace to this island by peaceful means, something he worked towards throughout his political career, and was vehemently opposed to political violence.”
Taoiseach Micheal Martin described Mr Currie as a “peacemaker”.
Mr Martin tweeted: “Saddened to hear of the death of Austin Currie, one of the founding fathers of the civil rights movement in Northern Ireland.
“He did so much for people, as a peacemaker and in politics, serving in the Dail and as Minister of State with distinction.
“My sympathies to his family.”
Irish Foreign Affairs Minister Simon Coveney tweeted: “So sad to learn of the passing of Austin Currie.
“A man of extraordinary generosity & conviction, he campaigned for social justice, equality and peace all his life, North and South.”